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hwk1-2011

# hwk1-2011 - STOR 890 Spring 2011 Homework 1 Note Try these...

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STOR 890, Spring 2011, Homework 1 Note: Try these problems yourselves and contribute to the class discussion on Tue. 2/22. (1) A combined spot/futures market contains a bank account B , a stock S and a futures contract FU related to S . Suppose the following data are collected: S (0) = 1 , S (1) = 1 . 02 , S (2) = 0 . 99 , S (3) = 1 . 01 , (unit price in dollar); FU (0 , 3) = 1 . 04 , FU (1 , 3) = 1 . 01 , FU (2 , 3) = 1 . 02 , (unit price in dollar); B (0) = 1 , r (1) = 0 . 001 , r (2) = r (3) = 0 . 002 . (1a) How much does it cost to take a long position of FU (with 100 shares) at t = 1? (1b) For a short position of FU (with 100 shares) taken at t = 2, what is the payoff at t = 3? (1c) Frank starts with \$1,000 in B . Suppose at t = 1, he shorts 100 shares of S and takes a long position of FU with 50 shares and the maturity date T = 3. What is the gain G ( t ) in Frank’s portfolio at t = 2 before marking to market? What is the value V ( t ) of his portfolio at t = 3? (2) Construct an example of arbitrage-free but incomplete single period model. (3) Consider the binomial tree model: T = 3, u = 1 . 07, d = 0 . 92, r = 6% and S (0) = 2 (see the example in Lecture 2). Suppose a constant dividend yield λ = 5% of the stock price is issued at
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